CN114741839A - FDTD method for analyzing transmission of very-low frequency electromagnetic wave in earth-ionized layer - Google Patents

FDTD method for analyzing transmission of very-low frequency electromagnetic wave in earth-ionized layer Download PDF

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CN114741839A
CN114741839A CN202210200191.0A CN202210200191A CN114741839A CN 114741839 A CN114741839 A CN 114741839A CN 202210200191 A CN202210200191 A CN 202210200191A CN 114741839 A CN114741839 A CN 114741839A
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郑奎松
闫增辉
张兴铭
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Northwestern Polytechnical University
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Abstract

The invention discloses an FDTD method for analyzing the propagation of very-low frequency electromagnetic waves in a ground-ionized layer. The method comprises the steps of firstly simulating the earth-ionosphere electromagnetic environment, then simplifying the operation through two times of coordinate system conversion, then calculating the specific propagation condition by using an SO-FDTD algorithm, and finally controlling the analysis of the earth-ionosphere parameters to summarize the propagation characteristics. The invention expands from the calculation of a simple radiation field to the propagation characteristic, further researches the propagation of the very low frequency electromagnetic wave in the ground-ionosphere waveguide, and has important reference significance for the research in the fields of ultra-long-distance navigation, submarine communication, weather prediction and the like. The invention can solve and analyze the problem about the propagation characteristics of the very low frequency electromagnetic wave in the earth-ionosphere waveguide.

Description

FDTD method for analyzing transmission of very-low frequency electromagnetic wave in earth-ionized layer
Technical Field
The invention belongs to the technical field of electromagnetic field numerical calculation, and particularly relates to an FDTD method for propagating electromagnetic waves in a ground-ionized layer.
Background
The very low frequency electromagnetic wave is an electromagnetic wave with the frequency of 3 kHz-30 kHz, has the advantages of long propagation distance, small propagation loss, stable amplitude and phase, strong permeability and the like, and is widely applied to the fields of ultra-long distance navigation, submarine communication, weather prediction and the like. The ground-ionosphere has good reflection characteristics for the very low frequency electromagnetic waves, and the wavelength of the very low frequency electromagnetic waves is close to the distance between the ground-ionosphere, so that the propagation of the very low frequency electromagnetic waves between the ground-ionosphere can be similar to the propagation in a waveguide, and is also called a ground-ionosphere waveguide propagation mode. However, in practice, the ground, the ionosphere and the electromagnetic space-time variation between the ground and the ionosphere are extremely complex, so that very low frequency electromagnetic waves exhibit very complex characteristics when propagating in the earth-ionosphere, and therefore, achieving more accurate prediction is of great significance to the above application fields.
When the very low frequency electromagnetic wave is calculated to propagate in the earth-ionosphere, the analytic method is not suitable in consideration of the need of further improving the accuracy of prediction. In the numerical method, FDTD (Finite Difference Time Domain) is generally used to simulate the propagation of the very low frequency electromagnetic wave in the earth-ionosphere waveguide. However, in most of the existing achievements, the field distribution situation of the very low frequency electromagnetic wave when propagating in the earth-ionosphere waveguide is calculated only by FDTD, and the current research is less for analyzing the propagation characteristics of the very low frequency electromagnetic wave in the earth-ionosphere waveguide by FDTD, especially for the influence of the various earth-ionosphere parameters on propagation, which is very important for improving the prediction speed and accuracy. Therefore, it is very necessary to design an algorithm capable of analyzing the propagation characteristics of very low frequency electromagnetic waves in the earth-ionosphere waveguide.
Disclosure of Invention
In order to overcome the disadvantages of the prior art, the present invention provides an FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere. The method comprises the steps of firstly simulating the earth-ionosphere electromagnetic environment, then simplifying the operation through two times of coordinate system conversion, then calculating the specific propagation condition by using an SO-FDTD algorithm, and finally controlling the analysis of the earth-ionosphere parameters to summarize the propagation characteristics. The invention expands from the calculation of a simple radiation field to the propagation characteristic, further researches the propagation of the very low frequency electromagnetic wave in the ground-ionosphere waveguide, and has important reference significance for the research in the fields of ultra-long-distance navigation, submarine communication, weather prediction and the like. The invention can solve and analyze the problem of the propagation characteristic of the very low frequency electromagnetic wave in the earth-ionosphere waveguide.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: taking the earth center as a pole, and taking a ray obtained by extending the connection line of the earth center and the electromagnetic wave emission point as a polar axis, establishing a polar coordinate system, and converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system;
data are crawled from an ionosphere IRI2016 model and an NRLMSISE-00 atmosphere model, and a real-time data set of the density and the temperature of particles on a propagation path of a very low frequency electromagnetic wave in a ground-ionosphere waveguide is established;
step 2: from the real-time data set obtained in step 1, the parameter, electron density N, is calculatedeAnd a collision frequency v, simulating the earth-ionosphere environment on the propagation path in real time;
and step 3: the electron density N under a polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system after being processed by a bilinear interpolation algorithm;
and 4, step 4: electron density N converted to rectangular coordinate system according to step 3eCalculating the field distribution condition of the very low frequency electromagnetic wave on the propagation path by using an SO-FDTD algorithm, and comparing and verifying the field intensity conversion condition of the electromagnetic wave receiving point at different times in one day with the actually measured data on the very low frequency station;
and 5: and controlling the parameter change of each region-ionosphere, and analyzing the field distribution condition obtained by calculation to obtain the propagation characteristic of the very low frequency electromagnetic wave in the region-ionosphere waveguide.
Further, the step 1 specifically includes the following steps:
step 1-1: electron density N on propagation path of IRI2016 model of ionosphere obtained by crawling of crawlereAnd electron temperature TeCrawling oxygen molecule density on NRLMSISE-00 atmospheric model propagation path
Figure BDA0003529047490000021
Density of oxygen atoms NoAnd nitrogen moleculesDensity of
Figure BDA0003529047490000022
；
Step 1-2: establishing a real-time data set of particle density and temperature on a propagation path of the very low frequency electromagnetic wave in the earth-ionosphere waveguide;
taking the earth center as a pole point, and taking a ray obtained by extending the connection line of the earth center and the electromagnetic wave transmitting point as a polar axis, establishing a polar coordinate system, and converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system; for any point P (rho, theta) in the two-dimensional polar coordinate system, the polar diameter rho corresponds to the distance from the point to the geocenter, and the polar angle theta represents the included angle between the connecting line of the point and the geocenter and the connecting line of the electromagnetic wave emission point and the geocenter; from an electromagnetic wave emitting point to an electromagnetic wave receiving point, crawling real-time data of particle density and temperature at a certain height from the ground to a low ionization layer at a corresponding position once by a crawler at intervals of an angle to establish a real-time data set;
further, the step 2 specifically includes the following steps:
step 2-1: simulating a ground environment;
the ground is approximated to be a uniform electromagnetic medium, as shown in table 1:
TABLE 1 terrestrial approximate electromagnetic Medium
Dielectric constant ε Magnetic permeability mu
Average land state 10 0.003
Average sea state 80 4
Step 2-2: simulating an ionospheric environment;
two parameters of the ionosphere-electron density NeAnd a collision frequency v, wherein the electron density NeDirectly obtaining from an ionosphere IRI2016 model, and calculating the collision frequency v by an empirical formula (1), which is as follows:
Figure BDA0003529047490000031
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003529047490000032
in the formula, Ve,iRepresenting the collision frequency of electrons and ions,
Figure BDA0003529047490000033
representing the collision frequency, V, of electrons and oxygen moleculese,ORepresenting the collision frequency of electrons and oxygen atoms,
Figure BDA0003529047490000034
representing the collision frequency of electrons and nitrogen molecules, NeRepresents the electron density, TeWhich is representative of the temperature of the electrons,
Figure BDA0003529047490000035
represents the molecular density of oxygen, NOAnd represents the density of oxygen atoms,
Figure BDA0003529047490000036
represents the molecular density of nitrogen;
step 2-3: simulating an earth-ionosphere environment;
for the space between the ground and the ionosphere, an empirical index model is used for representation, which is as follows:
v(z)＝1.82×1011e-0.15z (3)
N(z)＝1.43×107e-0.15He[(β-0.15)(z-H)] (4)
wherein v (z) represents the collision frequency at height z, N (z) represents the electron density at height z, and β represents the gradient coefficient (km)-1) H represents ionospheric reference height (km);
the recommended values of beta and H in the low and medium latitude areas in the empirical index model are shown in a table 2, wherein f is the working frequency;
TABLE 2 values of beta and H in low and medium latitude areas
(Summer) Winter season
Daytime β＝0.3,H＝70 β＝0.3,H＝72
At night β＝0.0077f+0.31,H＝87 β＝0.0077f+0.31,H＝87
Further, the step 3 specifically includes the following steps:
for FDTDSince Yee cells in the polar coordinate system and the rectangular coordinate system are not in one-to-one correspondence, direct conversion cannot be achieved, and processing needs to be performed through interpolation; applying bilinear interpolation algorithm to convert electron density N in polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system; the core idea of the bilinear interpolation algorithm is that interpolation is respectively carried out in the x direction and the y direction for one time, namely the value of each point in a rectangular coordinate system is obtained by interpolation of points in four adjacent polar coordinate systems; for the value of point P (x, y), Q is known11(x1,y1)，Q12(x2,y2)，Q13(x3,y3) And Q14(x4,y4) The values of four points and the interpolation formula are
Figure BDA0003529047490000041
Further, the step 4 specifically includes the following steps:
step 4-1: calculating field distribution on a propagation path by using SO-FDTD;
the maxwell rotation equation for a linear isotropic medium is:
Figure BDA0003529047490000042
wherein G represents magnetic field strength, D represents electric flux density, sigma represents electric conductivity, E represents electric field strength, B represents magnetic flux density, sigmamRepresents equivalent permeability;
the dielectric constant of the ionosphere varies with frequency, so the ionosphere also belongs to a dispersive medium, for which there are:
B＝μG (7)
D(ω)＝ε(ω)E(ω) (8)
wherein μ represents permeability, ε (ω) represents permittivity, and ω represents angular frequency;
FDTD discretization by formula (6) gives:
Figure BDA0003529047490000043
in the formula, n represents the number of iteration steps;
converting equation (8) from the frequency domain to the time domain, and for the x component, obtaining:
Figure BDA0003529047490000051
in the formula, epsilon0Represents the vacuum dielectric constant,. epsilonrRepresents a relative dielectric constant, Ex(t) represents the x component of E (t), Dx(t) represents the x component of D (t).
The ionosphere is the Drude medium, so there are:
Figure BDA0003529047490000052
in the formula, plAnd q islIs a polynomial coefficient, and M and N represent the total number of terms of the polynomial.
Namely, the method comprises the following steps:
Figure BDA0003529047490000053
setting a function:
Figure BDA0003529047490000054
the left end mean value is approximate, and the right end center difference is approximate:
Figure BDA0003529047490000055
introducing a discrete Shift operator zlDefined as:
zlfn＝fn+1 (15)
combining formula (15) with formula (14) to obtain:
Figure BDA0003529047490000056
combining the vertical type (13) and the formula (16) to obtain:
Figure BDA0003529047490000057
formula (17) is substituted for formula (12) to give:
Figure BDA0003529047490000058
two sides of the same ride (z)l+1)NObtaining:
Figure BDA0003529047490000061
because of the effect of the shift operator
Figure BDA0003529047490000062
The step equation is therefore obtained as:
Figure BDA0003529047490000063
in the formula, alAnd blBy the coefficient p in rational formula (11)0,p1,...,pNAnd q is0,q1,...,qMRepresents;
when M is equal to N is equal to 2, the finishing is as follows:
Figure BDA0003529047490000064
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003529047490000065
the ionosphere is a plasma, so there are:
Figure BDA0003529047490000066
wherein ω ispIs the plasma frequency, vcIs the electron impact frequency, ε∞＝1；
The step calculation procedure for SO-FDTD is summarized as follows: (1) calculated by E → G, using formula (9) in the first formula; (2) calculated by G → D using the second formula of formula (9); (3) calculated by D → E, using equation (20); therefore, the radiation field distribution situation generated by the propagation of the very low frequency electromagnetic wave in the earth-ionosphere can be gradually solved on the time-space axis;
step 4-2: and comparing the calculated data with the actually measured data to verify the feasibility of the method.
Further, the step 5 specifically includes the following steps:
step 5-1: setting a rectangular coordinate value as a variable, and solving a matrix numerical value of electromagnetic field distribution; drawing to obtain the amplitude value and phase value distribution of each electromagnetic field component on a propagation path, and analyzing the propagation characteristics of the very low frequency electromagnetic wave in the earth-ionosphere waveguide on the basis of the amplitude value and the phase value distribution;
step 5-2: considering the influence of each local-ionosphere parameter on the propagation process, setting each local-ionosphere parameter as a variable, and obtaining a conclusion of the propagation characteristic of the very low frequency electromagnetic wave in the local-ionosphere waveguide by comparing the amplitude and phase value distribution conditions of the electromagnetic field component when each local-ionosphere parameter is different.
The invention has the following beneficial effects:
compared with the existing very low frequency propagation calculation method in the earth-ionosphere waveguide, the method converts the problem in the three-dimensional spherical coordinate system into the two-dimensional polar coordinate system and then into the two-dimensional rectangular coordinate system, greatly reduces the calculation amount and has higher calculation efficiency. The method carries out simulation calculation on the distribution of a radiation field generated by the transmission of the very low frequency electromagnetic wave in the earth-ionosphere waveguide, expands the calculation of the radiation field into the research on the transmission characteristic of the very low frequency in the earth-ionosphere waveguide, and particularly explores the influence of the earth-ionosphere parameter on the transmission by controlling various earth-ionosphere parameters. The invention has novel thought and innovativeness and has important significance for the application aspect of the very low frequency electromagnetic wave.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic diagram of a problem model of the method of the present invention.
FIG. 3 is a schematic representation of a crawler crawling data by the method of the present invention.
FIG. 4 is a schematic diagram of a method for simulating a geoionospheric electromagnetic environment in real time according to the method of the present invention.
FIG. 5 is a schematic diagram of the coordinate system transformation according to the method of the present invention.
FIG. 6 is a schematic diagram of a method for converting a three-dimensional spherical coordinate system to a two-dimensional polar coordinate system according to the present invention.
FIG. 7 is a schematic diagram of a method for converting a two-dimensional polar coordinate system to a two-dimensional rectangular coordinate system according to the present invention.
FIG. 8 is a schematic representation of the SO-FDTD operating principle of the process of the present invention.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the drawings.
The technical problem solved by the invention is as follows: in order to improve the speed and the precision of calculating the space-time distribution of a radiation field generated by the propagation of the very low frequency electromagnetic waves in the earth-ionosphere waveguide in the prior art, the invention designs an FDTD method for analyzing the propagation of the very low frequency electromagnetic waves in the earth-ionosphere.
The invention is based on Visual studio compiling platform, uses Fortran language to simulate the ground-ionized layer electromagnetic environment in real time, firstly obtains the density and temperature of various particles from the ionized layer IRI2016 model and the NRLMSISE-00 atmosphere model, and then obtains two ionized layer parameters which directly act on the transmission, namely the electron density N through calculationeAnd collision frequency v, thus simulating the earth-ionosphere environment in real time. Then using SO-FDTD algorithmAnd calculating the distribution condition of the radiation field generated by the very low frequency electromagnetic wave in the earth-ionosphere waveguide, and comparing the field intensity change conditions of the receiving points in different time with actual data obtained by the very low frequency station to verify the feasibility of the algorithm. And finally, the influence of the parameters on the transmission of the very low frequency electromagnetic waves in the earth-ionosphere waveguide is specifically researched by controlling the parameter change of each earth-ionosphere, and the transmission characteristics of the very low frequency electromagnetic waves in the earth-ionosphere waveguide are summarized.
An FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere, comprising the steps of:
step 1: taking the earth center as a pole, and taking a ray obtained by extending the connection line of the earth center and the electromagnetic wave emission point as a polar axis, establishing a polar coordinate system, and converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system;
the method comprises the steps of utilizing a Requests library in a Python programming language to realize large-scale crawling of data from an ionosphere IRI2016 model (https:// ccmc. gsfc. nasa. gov/model web/models/IRI2016_ vitmo. php) and an NRLMSISE-00 atmosphere model (https:// ccmc. gsfc. nasa. gov/model web/models/NRLMSISE00.php), and establishing a real-time data set of the density and the temperature of various particles on a propagation path of the very-low-frequency electromagnetic wave in a ground-ionosphere waveguide;
step 2: calculating two parameters, namely electron density N, which have direct influence on the ionized layer by using an empirical formula according to the real-time data set obtained in the step 1eAnd a collision frequency v, simulating the earth-ionosphere environment on the propagation path in real time;
and step 3: the electron density N under a polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system after being processed by a bilinear interpolation algorithm;
and 4, step 4: electron density N converted to rectangular coordinate System according to step 3eCalculating the field distribution condition of the very low frequency electromagnetic wave on the propagation path by using an SO-FDTD algorithm, and comparing and verifying the field intensity conversion condition of the electromagnetic wave receiving point at different times in one day with the actually measured data on the very low frequency station;
and 5: and controlling the parameter change of each local-ionosphere, analyzing and summarizing the field distribution condition obtained by calculation, and obtaining the propagation characteristic of the very low frequency electromagnetic wave in the ground-ionosphere waveguide.
Further, the step 1 specifically includes the following steps:
step 1-1: crawlers are implemented using Requests libraries in the Python programming language to crawl the required data. For the ionosphere IRI2016 model (https:// ccmc. gsfc. nasa. gov/model web/models/IRI2016_ vitmo. php), it is necessary to crawl the electron density N on the propagation patheAnd electron temperature Te. For the NRLMSISE-00 atmosphere model (https:// ccmc. gsfc. nasa. gov/model web/models/nrlmsse 00.php), the oxygen molecular density on the propagation path needs to be crawled
Figure BDA0003529047490000081
Density of oxygen atoms NoAnd molecular density of nitrogen
Figure BDA0003529047490000082
。
Step 1-2: establishing a real-time data set of particle density and temperature on a propagation path of the very low frequency electromagnetic wave in the earth-ionosphere waveguide;
and establishing a polar coordinate system by taking the earth center as a pole and the ray obtained by extending the connecting line of the earth center and the transmitting point as a polar axis, thereby converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system. For any point P (rho, theta) in the coordinate system, the polar diameter rho corresponds to the distance from the point to the geocenter, and the polar angle theta represents the included angle between the connecting line of the point and the geocenter and the connecting line of the emission point and the geocenter. As shown in fig. 6, from the beginning of the emission point to the end of the receiving point (actually, considering the boundary in FDTD, so from the slightly outside of the emission point to the slightly outside of the receiving point), every very small angle, the crawler crawls the data set of the various particle densities and temperatures corresponding to every very small height from the ground to the low ionosphere, so that the data set of the various particle densities and temperatures in the propagation path is stored by using only one two-dimensional array.
Further, the step 2 specifically includes the following steps:
step 2-1: simulating a ground environment;
the ground is approximated to be a uniform electromagnetic medium, as shown in table 1:
TABLE 1 ground approximate electromagnetic Medium table
Dielectric constant ε Magnetic permeability mu
Average land state 10 0.003
Average sea state 80 4
Step 2-2: simulating an ionospheric environment;
for the ionosphere, there are two parameters that directly affect the effect of electron density NeAnd a collision frequency v, wherein the electron density NeDirectly obtaining from an ionosphere IRI2016 model, and calculating the collision frequency v by an empirical formula (1), which is as follows:
Figure BDA0003529047490000091
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003529047490000092
in the formula, Ve,iRepresenting the collision frequency of electrons and ions,
Figure BDA0003529047490000093
representing the collision frequency, V, of electrons and oxygen moleculese,ORepresenting the collision frequency of electrons and oxygen atoms,
Figure BDA0003529047490000094
representing the collision frequency of electrons and nitrogen molecules, NeRepresents the electron density, TeWhich is representative of the temperature of the electrons,
Figure BDA0003529047490000095
represents the molecular density of oxygen, NOAnd represents the density of oxygen atoms,
Figure BDA0003529047490000096
represents the molecular density of nitrogen;
step 2-3: simulating an earth-ionosphere environment;
referring to fig. 4, for the space between the ground and the ionosphere, since the height of the lower ionosphere boundary is about 65km in the daytime, the height of the lower ionosphere boundary is about 80km in the evening, and the space from the ground to the lower ionosphere boundary is vacant, an empirical index model is used for representation, which is specifically as follows:
v(z)＝1.82×1011e-0.15z (3)
N(z)＝1.43×107e-0.15He[(β-0.15)(z-H)] (4)
wherein v (z) represents the collision frequency at height z, N (z) represents the electron density at height z, and β represents the gradient coefficient (km)-1) H represents ionospheric reference height (km);
the recommended values of beta and H in the low and medium latitude areas in the empirical index model are shown in a table 2, wherein f is the working frequency;
TABLE 2 values of beta and H in low and medium latitude areas
(Summer) Winter season
Daytime β＝0.3,H＝70 β＝0.3,H＝72
At night β＝0.0077f+0.31,H＝87 β＝0.0077f+0.31,H＝87
Further, the step 3 specifically includes the following steps:
for FDTD, Yee cells in a polar coordinate system and a rectangular coordinate system are not in one-to-one correspondence, so that direct conversion cannot be realized, and interpolation is needed for processing; adopting bilinear interpolation algorithm to convert the electron density N under a polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system; the core idea of the bilinear interpolation algorithm is that interpolation is carried out in the x direction and the y direction respectively, namely the value of each point in the rectangular coordinate system is obtained by interpolation of points in four adjacent polar coordinate systems; for the value of point P (x, y), Q is known11(x1,y1)，Q12(x2,y2)，Q13(x3,y3) And Q14(x4,y4) The values of four points and the interpolation formula are
Figure BDA0003529047490000101
Further, the step 4 specifically includes the following steps:
step 4-1: calculating field distribution on a propagation path by using SO-FDTD;
the maxwell rotation equation of the linear isotropic medium is as follows:
Figure BDA0003529047490000111
wherein G represents magnetic field strength, D represents electric flux density, sigma represents electric conductivity, E represents electric field strength, B represents magnetic flux density, sigmamRepresents equivalent permeability;
the dielectric constant of the ionosphere varies with frequency, so the ionosphere also belongs to a dispersive medium, for which there are:
B＝μG (7)
D(ω)＝ε(ω)E(ω) (8)
wherein μ represents permeability, ε (ω) represents permittivity, and ω represents angular frequency;
FDTD discretization by formula (6) gives:
Figure BDA0003529047490000112
converting equation (8) from the frequency domain to the time domain, and obtaining, for the x component:
Figure BDA0003529047490000113
the ionosphere is a very typical Drude medium, so there are:
Figure BDA0003529047490000114
namely, the method comprises the following steps:
Figure BDA0003529047490000115
setting a function:
Figure BDA0003529047490000116
the left end mean value is approximate, and the right end center difference is approximate:
Figure BDA0003529047490000117
introducing a discrete Shift operator zlDefined as:
zlfn＝fn+1 (15)
combining formula (15) with formula (14) to obtain:
Figure BDA0003529047490000121
combining the vertical type (13) and the formula (16) to obtain:
Figure BDA0003529047490000122
formula (17) is substituted for formula (12) to give:
Figure BDA0003529047490000123
two sides of the same ride (z)l+1)NObtaining:
Figure BDA0003529047490000124
because of the effect of the shift operator
Figure BDA0003529047490000125
Thereby obtaining a step-by-step maleThe formula is as follows:
Figure BDA0003529047490000126
when M is equal to N is equal to 2, finishing to obtain:
Figure BDA0003529047490000127
wherein the content of the first and second substances,
Figure BDA0003529047490000128
the ionosphere is a plasma, so there are:
Figure BDA0003529047490000131
wherein ω ispIs the plasma frequency, vcIs the electron impact frequency, ε∞＝1；
The step calculation procedure for SO-FDTD is summarized as follows: (1) calculated by E → G, using formula (9) in the first formula; (2) calculated by G → D using the second formula of formula (9); (3) calculated by D → E, using equation (20); therefore, the radiation field distribution situation generated by the propagation of the very low frequency electromagnetic wave in the earth-ionosphere can be gradually solved on a time-space axis;
step 4-2: and comparing the calculated data with the actually measured data to verify the feasibility of the method.
Further, the step 5 specifically includes the following steps:
step 5-1: in order to conveniently analyze the propagation characteristics of the very low frequency electromagnetic waves in the ground-ionosphere waveguide, a Visual studio platform is programmed by using Fortran language, and rectangular coordinate values under a uniform coordinate system are set as variables to obtain matrix values of electromagnetic field distribution. And (4) utilizing Origin software to carry out mapping to obtain the amplitude value and phase value distribution of each electromagnetic field component on the propagation path, and analyzing the propagation characteristics of the very low frequency electromagnetic wave in the earth-ionosphere waveguide on the basis of the amplitude value and the phase value distribution.
Step 5-2: considering the influence of the local-ionosphere parameters on the propagation process, the local-ionosphere parameters can be set as variables, and the propagation characteristic conclusion of the very-low frequency electromagnetic wave in the local-ionosphere waveguide can be obtained by comparing the amplitude and phase value distribution conditions of the electromagnetic field component when the local-ionosphere parameters are different.
The specific embodiment is as follows:
the method comprises the steps of processing the initial data obtained by crawling through a Visual studio platform through Fortran language programming to construct a ground-ionosphere environment, simplifying the problem through two times of coordinate system transformation, calculating the field distribution condition on a propagation path by using an SO-FDTD algorithm, and controlling ground-ionosphere parameter analysis to summarize the propagation characteristics.
Fig. 1 shows a calculation flow chart. Firstly, simulating a real electromagnetic environment of the earth-ionosphere, for the ionosphere, crawling the density and temperature of various particles from an ionosphere IRI2016 model and an NRLMSISE-00 atmosphere model, and obtaining the electron density and the collision frequency by using an empirical formula; on the basis, the problem under the three-dimensional spherical coordinate system is converted into a two-dimensional polar coordinate system, and then the problem under the two-dimensional polar coordinate system is converted into a two-dimensional rectangular coordinate system by utilizing a bilinear interpolation algorithm; the SO-FDTD algorithm can calculate the field distribution condition on the propagation path, compare the calculation result with the measured data, verify the feasibility of the method; and controlling the parameters of the ionosphere of each region, observing the influence of the parameters on the propagation process, and analyzing and summarizing the propagation characteristics.
Fig. 2 shows a problem diagram. The ground, the space between the ground and an ionized layer and the ionized layer are arranged from inside to outside in sequence, and the transmitting point and the receiving point are both positioned on the ground. In actual propagation, the propagation of very low frequency electromagnetic waves in the earth-ionosphere can be approximated by the propagation in a waveguide.
Fig. 3 shows a schematic view of a crawler. The crawler sends a signal carrying time, longitude and latitude, height and other parameters to the ionosphere IRI2016 modelGET request, return electron density NeAnd electron temperature TeThe information of (1). Similarly, the crawler sends a GET request carrying parameters such as time, longitude and latitude, height and the like to the NRLMSISE-00 atmospheric model and returns the molecular density of the oxygen
Figure BDA0003529047490000141
Density of oxygen atoms NoAnd molecular density of nitrogen
Figure BDA0003529047490000142
And (4) information.
Fig. 4 is a schematic diagram of a method for simulating a ground-ionosphere electromagnetic environment in real time. The method comprises the steps of sequentially obtaining the ground, the space between the ground and an ionized layer and the ionized layer from inside to outside, obtaining the density and the temperature of various particles from an ionized layer IRI2016 model and an NRLMSISE-00 atmosphere model for the ionized layer, obtaining the electron density and the collision frequency by using an empirical formula, approximating a certain uniform electromagnetic medium for the ground, and approximating the electron density and the collision frequency by using an exponential empirical model for the space between the ground and the ionized layer, so that the ground-ionized layer electromagnetic environment can be simulated in real time.
Fig. 5 is a schematic diagram illustrating the coordinate system conversion. From left to right, a schematic diagram of a three-dimensional spherical coordinate system, a schematic diagram of a two-dimensional polar coordinate system and a schematic diagram of a two-dimensional rectangular coordinate system are sequentially arranged. The propagation of very low frequency electromagnetic waves in the earth-ionosphere waveguide is a problem in a three-dimensional spherical coordinate system. When the earth center is used as a pole, the ray obtained by extending the connecting line of the earth center and the transmitting point is used as a polar axis, a polar coordinate system is established, and the problem under the three-dimensional spherical coordinate system can be converted into a two-dimensional polar coordinate system. A rectangle which can contain a transmitting point and a receiving point simultaneously is taken in a polar coordinate system, and the problem under a two-dimensional polar coordinate system can be converted into a two-dimensional rectangular coordinate system.
Fig. 6 is a schematic diagram of a method for converting a three-dimensional spherical coordinate system into a two-dimensional polar coordinate system (or a schematic diagram of an initial data set construction). The left is a schematic diagram of a two-dimensional polar coordinate system, and the right is a schematic diagram of an initial data set. From the start of the emission point to the end of the reception point (actually, considering the boundary in FDTD, the boundary is from the outer part of the emission point to the outer part of the reception point), every other small angle, the variation of the various particle densities and temperatures of every other small height from the ground to the low ionosphere of the corresponding position is crawled once by a crawler, and thus, the data sets of the various particle densities and temperatures on the propagation path are stored by only one two-dimensional array.
Fig. 7 is a schematic diagram illustrating a method for converting a two-dimensional polar coordinate system to a two-dimensional rectangular coordinate system. Point P (x, y) corresponds to a certain Yee cell on the rectangular coordinate system of FDTD, point Q11(x1,y1)，Q12(x2,y2)，Q13(x3,y3) And Q14(x4,y4) Corresponding to four Yee cells on the FDTD polar coordinate nearest to the point P (x, y), the value of the point P (x, y) is represented by Q according to the bilinear interpolation algorithm11(x1,y1)，Q12(x2,y2)，Q13(x3,y3) And Q14(x4,y4) The values of the four points are determined, and the electron density and the collision frequency in a polar coordinate system are converted into a rectangular coordinate system.
FIG. 8 is a schematic diagram showing the operation principle of SO-FDTD. In the figure, each electric field component is surrounded by four magnetic field components, and each magnetic field component is surrounded by four electric field components; the electric field and the magnetic field are alternately sampled in time sequence, and the sampling time interval is different by half time step; e → H → D → E has a definite formula in each step, so that the radiation field distribution generated by the propagation of the very low frequency electromagnetic wave in the earth-ionosphere can be solved on the time-space axis by iterative stepwise advancing.

Claims (6)

1. An FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere, comprising the steps of:
step 1: taking the earth center as a pole, and taking a ray obtained by extending the connection line of the earth center and the electromagnetic wave emission point as a polar axis, establishing a polar coordinate system, and converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system;
crawling data from an ionosphere IRI2016 model and an NRLMSISE-00 atmosphere model, and establishing a real-time data set of the density and the temperature of particles on a propagation path of a very-low-frequency electromagnetic wave in a ground-ionosphere waveguide;
step 2: from the real-time data set obtained in step 1, the parameter, electron density N, is calculatedeAnd a collision frequency v, simulating the earth-ionosphere environment on the propagation path in real time;
and step 3: the electron density N under a polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system after being processed by a bilinear interpolation algorithm;
and 4, step 4: electron density N converted to rectangular coordinate system according to step 3eCalculating the field distribution condition of the very low frequency electromagnetic wave on the propagation path by using an SO-FDTD algorithm, and comparing and verifying the field intensity conversion condition of the electromagnetic wave receiving point at different times in one day with the actually measured data on the very low frequency station;
and 5: and controlling the parameter change of each region-ionosphere, and analyzing the field distribution condition obtained by calculation to obtain the propagation characteristic of the very low frequency electromagnetic wave in the region-ionosphere waveguide.
2. The FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere according to claim 1, wherein the step 1 specifically comprises the following processes:
step 1-1: electron density N on propagation path of IRI2016 model of ionosphere obtained by crawling of crawlereAnd electron temperature TeCrawling oxygen molecule density on NRLMSISE-00 atmospheric model propagation path
Figure FDA0003529047480000011
Density of oxygen atoms NoAnd molecular density of nitrogen
Figure FDA0003529047480000012
Step 1-2: establishing a real-time data set of particle density and temperature on a propagation path of the very low frequency electromagnetic wave in the earth-ionosphere waveguide;
taking the earth center as a pole point, and taking a ray obtained by extending the connection line of the earth center and the electromagnetic wave transmitting point as a polar axis, establishing a polar coordinate system, and converting the problem under the three-dimensional spherical coordinate system into a two-dimensional polar coordinate system; for any point P (rho, theta) in the two-dimensional polar coordinate system, the polar diameter rho corresponds to the distance from the point to the geocenter, and the polar angle theta represents the included angle between the connecting line of the point and the geocenter and the connecting line of the electromagnetic wave emission point and the geocenter; and (3) crawling real-time data of particle density and temperature at a certain height from the ground to a low ionization layer at a corresponding position once by a crawler at an interval of an angle from an electromagnetic wave transmitting point to an electromagnetic wave receiving point, and establishing a real-time data set.
3. The FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere according to claim 2, wherein the step 2 specifically comprises the following processes:
step 2-1: simulating a ground environment;
the ground is approximated as a uniform electromagnetic medium, as shown in table 1:
TABLE 1 ground approximate electromagnetic Medium table
Dielectric constant ε Magnetic permeability mu Average land state 10 0.003 Average sea state 80 4
Step 2-2: simulating an ionospheric environment;
two parameters of the ionosphere-electron density NeAnd a collision frequency v, wherein the electron density NeDirectly obtaining from an ionosphere IRI2016 model, and calculating the collision frequency v by an empirical formula (1), which is as follows:
Figure FDA0003529047480000021
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003529047480000022
in the formula, Ve,iRepresenting the collision frequency of electrons and ions,
Figure FDA0003529047480000023
representing the collision frequency, V, of electrons and oxygen moleculese,ORepresenting the collision frequency of electrons and oxygen atoms,
Figure FDA0003529047480000024
representing the collision frequency of electrons and nitrogen molecules, NeRepresents the electron density, TeWhich is representative of the temperature of the electrons,
Figure FDA0003529047480000025
represents the molecular density of oxygen, NOAnd represents the density of oxygen atoms,
Figure FDA0003529047480000026
represents the molecular density of nitrogen;
step 2-3: simulating an earth-ionosphere environment;
for the space between the ground and the ionosphere, an empirical index model is used for representation, which is as follows:
v(z)＝1.82×1011e-0.15z (3)
N(z)＝1.43×107e-0.15He[(β-0.15)(z-H)] (4)
wherein v (z) represents the collision frequency at height z, N (z) represents the electron density at height z, and β represents the gradient coefficient (km)-1) H represents ionospheric reference height (km);
the recommended values of beta and H in the low and medium latitude areas in the empirical index model are shown in a table 2, wherein f is the working frequency;
TABLE 2 values of beta and H in low and medium latitude areas
(Summer) Winter season Daytime β＝0.3,H＝70 β＝0.3,H＝72 At night β＝0.0077f+0.31,H＝87 β＝0.0077f+0.31,H＝87
4. An FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the earth-ionosphere according to claim 3, wherein the step 3 specifically comprises the following processes:
for FDTD, Yee cells in a polar coordinate system and a rectangular coordinate system are not in one-to-one correspondence, so that direct conversion cannot be realized, and interpolation is needed for processing; adopting bilinear interpolation algorithm to convert the electron density N under a polar coordinate systemeAnd the collision frequency v is converted into a rectangular coordinate system; the core idea of the bilinear interpolation algorithm is that interpolation is carried out in the x direction and the y direction respectively, namely the value of each point in the rectangular coordinate system is obtained by interpolation of points in four adjacent polar coordinate systems; for the value of point P (x, y), Q is known11(x1,y1)，Q12(x2,y2)，Q13(x3,y3) And Q14(x4,y4) The values of four points are interpolated by the formula
Figure FDA0003529047480000031
5. The FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the Earth-electric ionosphere according to claim 4, wherein the step 4 comprises the following steps:
step 4-1: calculating field distribution on a propagation path by using SO-FDTD;
the maxwell rotation equation of the linear isotropic medium is as follows:
Figure FDA0003529047480000032
wherein G represents magnetic field strength, D represents electric flux density, sigma represents electric conductivity, E represents electric field strength, B represents magnetic flux density, sigmamRepresents equivalent permeability;
the dielectric constant of the ionosphere varies with frequency, so the ionosphere also belongs to a dispersive medium, for which there are:
B＝μG (7)
D(ω)＝ε(ω)E(ω) (8)
wherein μ represents permeability, ε (ω) represents permittivity, and ω represents angular frequency;
FDTD discretization by formula (6) gives:
Figure FDA0003529047480000041
in the formula, n represents the number of iteration steps;
converting equation (8) from the frequency domain to the time domain, and for the x component, obtaining:
Figure FDA0003529047480000042
in the formula, epsilon0Represents the vacuum dielectric constant,. epsilonrRepresents a relative dielectric constant, Ex(t) represents the x component of E (t), Dx(t) represents the x component of D (t);
the ionosphere is the Drude medium, so there are:
Figure FDA0003529047480000043
in the formula, plAnd q islIs a polynomial coefficient, M and N represent the total number of polynomials;
namely, the method comprises the following steps:
Figure FDA0003529047480000044
setting a function:
Figure FDA0003529047480000045
the left end mean value is approximate, and the right end center difference is approximate:
Figure FDA0003529047480000046
introducing a discrete Shift operator zlDefined as:
zlfn＝fn+1 (15)
combining formula (15) with formula (14) to obtain:
Figure FDA0003529047480000047
combining the vertical type (13) and the formula (16) to obtain:
Figure FDA0003529047480000051
formula (17) is substituted for formula (12) to give:
Figure FDA0003529047480000052
two sides of the same ride (z)l+1)NObtaining:
Figure FDA0003529047480000053
because of the effect of the shift operator
Figure FDA0003529047480000054
The step equation is therefore obtained as:
Figure FDA0003529047480000055
in the formula, alAnd blBy the coefficient p in rational formula (11)0,p1,...,pNAnd q is0,q1,...,qMRepresents;
when M is equal to N is equal to 2, finishing to obtain:
Figure FDA0003529047480000056
wherein the content of the first and second substances,
Figure FDA0003529047480000057
the ionosphere is a plasma, so there are:
Figure FDA0003529047480000058
wherein ω ispIs the plasma frequency, vcIs the electron impact frequency, ε∞＝1；
The step calculation procedure for SO-FDTD is summarized as follows: (1) calculated by E → G, using formula (9) in the first formula; (2) calculated by G → D using the second formula of formula (9); (3) calculated by D → E, using equation (20); therefore, the radiation field distribution situation generated by the propagation of the very low frequency electromagnetic wave in the earth-ionosphere can be gradually solved on the time-space axis;
step 4-2: and comparing the calculated data with the actually measured data to verify the feasibility of the method.
6. The FDTD method for analyzing the propagation of very low frequency electromagnetic waves in the Earth-electric ionosphere according to claim 5, wherein the step 5 comprises the following steps:
step 5-1: setting a rectangular coordinate value as a variable, and solving a matrix numerical value of electromagnetic field distribution; plotting to obtain the amplitude value and phase value distribution of each electromagnetic field component on the propagation path, and analyzing the propagation characteristic of the very low frequency electromagnetic wave in the earth-ionosphere waveguide based on the amplitude value and the phase value distribution;
step 5-2: considering the influence of each local-ionosphere parameter on the propagation process, setting each local-ionosphere parameter as a variable, and obtaining a conclusion of the propagation characteristic of the very low frequency electromagnetic wave in the local-ionosphere waveguide by comparing the amplitude and phase value distribution conditions of the electromagnetic field component when each local-ionosphere parameter is different.
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